Geodesic
Stability of the class of divisible $S$-acts
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 726-731.

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We describe monoids $S$ such that the theory of the class of all divisible $S$-acts is stable, superstable or, for commutative monoid, $\omega$-stable. More precisely, we prove that the theory of the class of all divisible $S$-acts is stable (superstable) iff $S$ is a linearly ordered (well ordered) monoid. We also prove that for a commutative monoid $S$ the theory of the class of all divisible $S$-acts is $\omega$-stable iff $S$ is either an abelian group with at most countable number of subgroups or is finite and has only one proper ideal. Classes of regular, projective and strongly flat $S$-acts were considered in [1, 2]. Using results from [3] we obtain necessary and sufficient conditions for stability, superstability and $\omega$-stability of theories of classes of all divisible $S$-acts.
Keywords: monoid, stability, superstability, $\omega$-stability.
Mots-clés : divisible $S$-act 
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A. I. Krasitskaya. Stability of the class of divisible $S$-acts. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 17 (2020), pp. 726-731. http://geodesic.mathdoc.fr/item/SEMR_2020_17_a13/

[1] A.V. Mikhalev, E.V. Ovchinnikova, E.A. Palyutin, A.A. Stepanova, “Model theoretic properties of regular polygons”, J. Math. Sci. (N.Y.), 140:2 (2007), 250–285 | DOI | MR

[2] V. Gould, A.V. Mikhalev, E.A. Palyutin, A.A. Stepanova, “Model theoretic properties of free, projective, and flat S-acts”, J. Math. Sci. New York, 164:2 (2010), 195–227 | DOI | MR | Zbl

[3] T.G. Mustafin, “On stability theory of polygons”, Trudy Inst. Mat., 8 (1988), 92–108 | MR | Zbl

[4] M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories, Walter De Gruyter, Berlin, 2000 | MR | Zbl

[5] A.I. Krasitskaya, A.A. Stepanova, Primitive normality and primitive connection of divisible acts, Algebra and Logic (to appear) | MR

[6] Yu.L. Ershov, E.A. Palyutin, Mathematical logic, Fizmathlit, M., 2011 | MR | Zbl

[7] C.C. Chang, H.J. Keisler, Model theory, Elsevier, New York, 1973 | MR | Zbl

[8] V.S. Bogomolov, T.G. Mustafin, “Description of commutative monoids over which all polygons are W-stable”, Algebra and Logic, New York, 28:4 (1989), 239–247 | MR | Zbl